Multi-speed transmission

ABSTRACT

A family of transmission gearing arrangements produces at least nine forward speed ratios and one reverse speed ratio by selective engagement of three shift elements in various combinations. Each disclosed transmission includes four planetary gear sets, three brakes, and either three or four clutches. One embodiment produces twelve forward speed ratios and two reverse speed ratios. Another embodiment produces thirteen forward speed ratios.

TECHNICAL FIELD

This disclosure relates to the field of automatic transmissions for motor vehicles. More particularly, the disclosure pertains to an arrangement of gears, clutches, and the interconnections among them in a power transmission.

BACKGROUND

Many vehicles are used over a wide range of vehicle speeds, including both forward and reverse movement. Some types of engines, however, are capable of operating efficiently only within a narrow range of speeds. Consequently, transmissions capable of efficiently transmitting power at a variety of speed ratios are frequently employed. When the vehicle is at low speed, the transmission is usually operated at a high speed ratio such that it multiplies the engine torque for improved acceleration. At high vehicle speed, operating the transmission at a low speed ratio permits an engine speed associated with quiet, fuel efficient cruising. Typically, a transmission has a housing mounted to the vehicle structure, an input shaft driven by an engine crankshaft, and an output shaft driving the vehicle wheels, often via a differential assembly which permits the left and right wheel to rotate at slightly different speeds as the vehicle turns.

SUMMARY OF THE DISCLOSURE

In a first embodiment, a transmission includes a first gearing arrangement, first and second simple planetary gear sets, a first brake, and a first clutch. The first gearing arrangement is configured to selectively constrain a speed ratio between a third shaft and an input shaft to various values include a negative value, zero, two values between zero and one, and one. The sun gear of the first planetary gear set is fixedly coupled to the third shaft while the carrier of the first planetary gear set is fixedly coupled to the input shaft. The ring gear of the second planetary gear set is fixedly coupled to the third shaft while the carrier of the second planetary gear set is fixedly coupled to the output shaft. The first clutch selectively couples the ring gear of the first gear set to the output shaft. The first brake selectively holds the sun gear of the second gear set against rotation. The first gearing arrangement may include a gearing arrangement that fixedly imposes a linear speed relationship among first, second, third, and fourth shafts, second and third clutches that selectively couple the first and second shaft, respectively, to the input shaft, and second and third brake that selectively hold the second and fourth shafts, respectively, against rotation. Alternatively, the first gearing arrangement may include a gearing arrangement that fixedly imposes a linear speed relationship among first, second, third, and fourth shafts, second and third clutches that selectively couple the first and fourth shaft, respectively, to the input shaft, and second and third brake that selectively hold the first and second shafts, respectively, against rotation.

In a second embodiment, a transmission includes first, second, and third gearing arrangements, two clutches, and three brakes. The first gearing arrangement fixedly imposes a linear speed relationship among first, second, third, and fourth shafts. The second gearing arrangement fixedly imposes a linear speed relationship among the third shaft, an input shaft, and a fifth shaft. The third gearing arrangement fixedly imposes a linear speed relationship among the third shaft, an output shaft, and a sixth shaft. First, second, and third brakes selectively hold the sixth, second, and fourth shafts, respectively, against rotation. A first clutch selectively couples the fifth shaft to the output shaft. A third clutch selectively couples second shaft to the input shaft. Second and fourth clutches may selectively couple the input shaft and fifth shaft, respectively, to the first shaft.

In a third embodiment, a transmission includes first, second, and third gearing arrangements, two clutches, and three brakes. The first gearing arrangement fixedly imposes a linear speed relationship among first, second, third, and fourth shafts. The second gearing arrangement fixedly imposes a linear speed relationship among the third shaft, an input shaft, and a fifth shaft. The third gearing arrangement fixedly imposes a linear speed relationship among the third shaft, an output shaft, and a sixth shaft. First, second, and third brakes selectively hold the sixth, first, and second shafts, respectively, against rotation. A first clutch selectively couples the fifth shaft to the output shaft. A second clutch selectively couples first shaft to the input shaft. Third and fourth clutches may selectively couple the input shaft and fifth shaft, respectively, to the fourth shaft.

In a fourth embodiment, a transmission includes first, second, and third gearing arrangements, two clutches, and three brakes. The first gearing arrangement fixedly imposes a linear speed relationship among second, third, and fourth shafts. The second gearing arrangement fixedly imposes a linear speed relationship among the third shaft, an input shaft, and a fifth shaft. The third gearing arrangement fixedly imposes a linear speed relationship among the third shaft, an output shaft, and a sixth shaft. First, second, and third brakes selectively hold the sixth, second, and fourth shafts, respectively, against rotation. A first clutch selectively couples the fifth shaft to the output shaft. A third clutch selectively couples second shaft to the input shaft. A fourth gearing arrangement may selectively impose a linear speed relationship among the input shaft, the third shaft, and the fourth shaft. Alternatively, a fourth gearing arrangement may selectively impose a linear speed relationship among the fifth shaft, the third shaft, and the fourth shaft.

In a fifth embodiment, a transmission includes first, second, and third gearing arrangements, two clutches, and two brakes. The first gearing arrangement selectively constrains a first shaft and a third shaft to rotate in opposite directions. The second gearing arrangement fixedly imposes a linear speed relationship among the third shaft, an input shaft, and a fifth shaft. The third gearing arrangement fixedly imposes a linear speed relationship among the third shaft, an output shaft, and a sixth shaft. First and second brakes selectively hold the sixth and first shafts, respectively, against rotation. A first clutch selectively couples the fifth shaft to the output shaft. A second clutch selectively couples first shaft to the input shaft. A fourth gearing arrangement may selectively impose a linear speed relationship among the input shaft, the third shaft, and the first shaft. Alternatively, a fourth gearing arrangement may selectively impose a linear speed relationship among the fifth shaft, the third shaft, and the first shaft.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a first transmission gearing arrangement.

FIG. 2 is a schematic diagram of a second transmission gearing arrangement.

FIG. 3 is a schematic diagram of a third transmission gearing arrangement.

FIG. 4 is a lever diagram corresponding to the third transmission gearing arrangement.

FIG. 5 is a schematic diagram of a fourth transmission gearing arrangement.

FIG. 6 is a schematic diagram of a fifth transmission gearing arrangement.

FIG. 7 is a schematic diagram of a sixth transmission gearing arrangement.

FIG. 8 is a lever diagram corresponding to the sixth transmission gearing arrangement.

DETAILED DESCRIPTION

Embodiments of the present disclosure are described herein. It is to be understood, however, that the disclosed embodiments are merely examples and other embodiments can take various and alternative forms. The figures are not necessarily to scale; some features could be exaggerated or minimized to show details of particular components. Therefore, specific structural and functional details disclosed herein are not to be interpreted as limiting, but merely as a representative basis for teaching one skilled in the art to variously employ the present invention. As those of ordinary skill in the art will understand, various features illustrated and described with reference to any one of the figures can be combined with features illustrated in one or more other figures to produce embodiments that are not explicitly illustrated or described. The combinations of features illustrated provide representative embodiments for typical applications. Various combinations and modifications of the features consistent with the teachings of this disclosure, however, could be desired for particular applications or implementations.

A gearing arrangement is a collection of rotating elements and shift elements configured to impose specified speed relationships among the rotating elements. Some speed relationships, called fixed speed relationships, are imposed regardless of the state of any shift elements. Other speed relationships, called selective speed relationships, are imposed only when particular shift elements are fully engaged. A linear speed relationship exists among an ordered list of rotating elements when i) the first and last rotating element in the group are constrained to have the most extreme speeds, ii) the speeds of the remaining rotating elements are each constrained to be a weighted average of the first and last rotating element, and iii) when the speeds of the rotating elements differ, they are constrained to be in the listed order, either increasing or decreasing. A discrete ratio transmission has a gearing arrangement that selectively imposes a variety of speed ratios between an input shaft and an output shaft.

A group of rotating elements are fixedly coupled to one another if they are constrained to rotate as a unit in all operating conditions. Rotating elements can be fixedly coupled by spline connections, welding, press fitting, machining from a common solid, or other means. Slight variations in rotational displacement between fixedly coupled elements can occur such as displacement due to lash or shaft compliance. One or more rotating elements that are all fixedly coupled to one another may be called a shaft. In contrast, two rotating elements are selectively coupled by a shift element when the shift element constrains them to rotate as a unit whenever it is fully engaged and they are free to rotate at distinct speeds in at least some other operating condition. A shift element that holds a rotating element against rotation by selectively connecting it to the housing is called a brake. A shift element that selectively couples two or more rotating elements to one another is called a clutch. Shift elements may be actively controlled devices such as hydraulically or electrically actuated clutches or brakes or may be passive devices such as one way clutches or brakes.

An example transmission is schematically illustrated in FIG. 1. The transmission utilizes four simple planetary gear sets 20, 30, 40, and 50. A planet carrier 22 rotates about a central axis and supports a set of planet gears 24 such that the planet gears rotate with respect to the planet carrier. External gear teeth on the planet gears mesh with external gear teeth on a sun gear 26 and with internal gear teeth on a ring gear 28. The sun gear and ring gear are supported to rotate about the same axis as the carrier. Gear sets 30, 40, and 50 are similarly structured.

A simple planetary gear set is a type of gearing arrangement that imposes a fixed linear speed relationship among the sun gear, the planet carrier, and the ring gear. Other known types of gearing arrangements also impose a fixed linear speed relationship among three rotating elements. For example, a double pinion planetary gear set imposes a fixed linear speed relationship between the sun gear, the ring gear, and the planet carrier.

A suggested ratio of gear teeth for each planetary gear set is listed in Table 1.

TABLE 1 Ring 28/Sun 26 1.50 Ring 38/Sun 36 2.00 Ring 48/Sun 46 1.50 Ring 58/Sun 56 1.50

In the transmission of FIG. 1, sun gear 26 is fixedly coupled to ring gear 38, carriers 22 and 32 are fixedly coupled to sun gear 46 and ring gear 58, carrier 42 is fixedly coupled to input shaft 10, and output shaft 12 is fixedly coupled to carrier 52. Ring gear 28 is selectively held against rotation by brake 66 and selectively coupled to input shaft 10 by clutch 64. Sun gear 26 and ring gear 38 are selectively held against rotation by brake 68. Input shaft 10 and carrier 42 are selectively coupled to sun gear 36 by clutch 70. Sun gear 56 is selectively held against rotation by brake 62. Finally, output shaft 12 and carrier 52 are selectively coupled to ring gear 48 by clutch 60. The combination of gear set 30 and clutch 70 selectively impose a linear speed relationship among ring gear 38, carrier 32, and input shaft 10.

As shown in Table 2, engaging the shift elements in combinations of three establishes nine forward speed ratios and one reverse speed ratio between input shaft 10 and output shaft 12. An X indicates that the shift element is required to establish the speed ratio. An (X) indicates the clutch can be applied but is not required. In 4th gear, clutch 60 and brake 62 establish the power flow path between input shaft 10 and output shaft 12. Any one of the remaining shift elements can also be applied. Applying clutch 64 ensures that all single and two step shifts from 4th gear can be accomplished by engaging only one shift element and releasing only one shift element. When the gear sets have tooth numbers as indicated in Table 1, the speed ratios have the values indicated in Table 2.

TABLE 2 60 62 64 66 68 70 Ratio Step Rev X X X −3.33 67% 1^(st) X X X 5.00 2^(nd) X X X 2.78 1.80 3^(rd) X X X 1.67 1.67 4^(th) X X (X) 1.27 1.32 5^(th) X X X 1.00 1.27 6^(th) X X X 0.79 1.27 7^(th) X X X 0.69 1.14 8^(th) X X X 0.60 1.15 9^(th) X X X 0.50 1.20

Gear sets 20 and 30, clutches 64 and 70, and brakes 66 and 68 form a gearing arrangement that produces a variety of speed ratios between input shaft 10 and carriers 22 and 32. Specifically, when clutch 70 and brake 66 are engaged, carriers 22 and 32 rotate in the opposite direction as input shaft 10. When brakes 66 and 68 are both engaged, carriers 22 and 32 are held against rotation. Engaging brake 68 in combination with clutch 70 or clutch 64 results in two underdrive ratios. Finally, when clutches 64 and 70 are both engaged carriers 22 and 32 rotate in unison with input shaft 10.

The transmission of FIG. 2 is like the transmission of FIG. 1 except that clutch 70 is deleted and sun gear 36 is selectively coupled to ring gear 48 by clutch 72. Suggested numbers of gear teeth for each planetary gear set is listed in Table 3.

TABLE 3 Ring 28/Sun 26 2.00 Ring 38/Sun 36 2.50 Ring 48/Sun 46 1.50 Ring 58/Sun 56 1.50

As shown in Table 4, engaging the clutches and brakes in combinations of three establishes nine forward speed ratios and one reverse speed ratio between input shaft 10 and output shaft 12. When the gear sets have tooth numbers as indicated in Table 3, the speed ratios have the values indicated in Table 4.

TABLE 4 60 62 64 66 68 72 Ratio Step Rev X X X −3.33 67% 1^(st) X X X 4.17 2^(nd) X X X 2.50 1.67 3^(rd) X X X 1.67 1.50 4^(th) X X (X) 1.27 1.32 5^(th) X X X 1.00 1.27 6^(th) X X X 0.82 1.22 7^(th) X X X 0.71 1.15 8^(th) X X X 0.60 1.19 9^(th) X X X 0.50 1.20

The transmission of FIG. 3 is like the transmissions of FIG. 1 and FIG. 2 except that both clutch 70 and 72 are present. Suggested numbers of gear teeth for each planetary gear set is listed in Table 5.

TABLE 5 Ring 28/Sun 26 2.20 Ring 38/Sun 36 1.80 Ring 48/Sun 46 1.50 Ring 58/Sun 56 1.50

As shown in Table 6, engaging the clutches and brakes in combinations of three establishes nine forward speed ratios and one reverse speed ratio between input shaft 10 and output shaft 12. When the gear sets have tooth numbers as indicated in Table 5, the speed ratios have the values indicated in Table 6.

TABLE 6 60 62 64 66 68 70 72 Ratio Step R1 X X X −4.93 106% R2 X X X −2.29 2.15  1^(st) X X X 4.67  2^(nd) X X X 3.47 1.35  3^(rd) X X X 2.42 1.43  4^(th) X X X 1.67 1.46  5^(th) X X X 1.27 1.32  6^(th) X X X 1.00 1.27  7^(th) X X X 0.83 1.21  8^(th) X X X 0.74 1.11  9^(th) X X X 0.70 1.06 10^(th) X X X 0.60 1.17 11^(th) X X X 0.53 1.14 12^(th) X X X 0.47 1.14

FIG. 4 shows a lever diagram corresponding to the gearing arrangements of FIGS. 1-3. Levers 74, 76, and 78 represent gearing arrangements that impose fixedly linear speed relationships. Lever 74 corresponds to gear sets 20 and 30 and represents the linear speed relationship among four shafts that is imposed by these two gear sets. The first shaft corresponds to sun gear 36. The second shaft corresponds to ring gear 28. The third shaft corresponds to the combination of carrier 22, carrier 32, sun gear 46, and ring gear 58. The fourth shaft corresponds to the combination of sun gear 26 and ring gear 38. There are many gearing arrangements that impose this speed relationship. For example, any pair of planetary gear sets wherein two rotating elements of the first gear set are respectively fixedly coupled to two rotating elements of the second gear set will impose a linear speed relationship among four shafts. Lever 76 corresponds to gear set 40. The fifth shaft corresponds to ring gear 48. Lever 78 corresponds to gear set 50. The sixth shaft corresponds to sun gear 56.

Another example transmission is schematically illustrated in FIG. 5. The transmission utilizes four simple planetary gear sets 80, 90, 40, and 50. A suggested ratio of gear teeth for each planetary gear set is listed in Table 7.

TABLE 7 Ring 88/Sun 86 1.90 Ring 98/Sun 96 2.00 Ring 48/Sun 46 1.50 Ring 58/Sun 56 1.50

In the transmission of FIG. 5, sun gear 86 is fixedly coupled to sun gear 96, ring gear 88 and carriers 92 are fixedly coupled to sun gear 46 and ring gear 58, carrier 42 is fixedly coupled to input shaft 10, and output shaft 12 is fixedly coupled to carrier 52. Sun gears 86 and 96 are selectively held against rotation by brake 102 and selectively coupled to input shaft 10 by clutch 100. Carrier 82 is selectively held against rotation by brake 104. Input shaft 10 and carrier 42 are selectively coupled to ring gear 98 by clutch 106. Sun gear 56 is selectively held against rotation by brake 62. Finally, output shaft 12 and carrier 52 are selectively coupled to ring gear 48 by clutch 60. The combination of gear set 80 and brake 104 selectively impose a speed relationship such that sun gear 86 and ring gear 88 rotate in opposite directions whenever brake 104 is engaged. The combination of gear set 90 and clutch 106 selectively impose a linear speed relationship among input shaft 10, carrier 92, and sun gear 96.

As shown in Table 8, engaging the shift elements in combinations of three establishes nine forward speed ratios and one reverse speed ratio between input shaft 10 and output shaft 12. An X indicates that the shift element is required to establish the speed ratio. An (X) indicates the clutch can be applied but is not required. In 4th gear, clutch 60 and brake 62 establish the power flow path between input shaft 10 and output shaft 12. Any one of the remaining shift elements can also be applied. Applying clutch 106 ensures that all single and two step shifts from 4th gear can be accomplished by engaging only one shift element and releasing only one shift element. When the gear sets have tooth numbers as indicated in Table 7, the speed ratios have the values indicated in Table 8.

TABLE 8 60 62 100 102 104 106 Ratio Step Rev X X X −3.17 78% 1^(st) X X X 4.08 2^(nd) X X X 2.50 1.63 3^(rd) X X X 1.67 1.50 4^(th) X X (X) 1.27 1.32 5^(th) X X X 1.00 1.27 6^(th) X X X 0.82 1.22 7^(th) X X X 0.72 1.14 8^(th) X X X 0.60 1.20 9^(th) X X X 0.50 1.21

Gear sets 80 and 90, clutches 100 and 106, and brakes 100 and 102 form a gearing arrangement that produces a variety of speed ratios between input shaft 10 and the combination of ring gear 88 and carrier 92. Specifically, when clutch 100 and brake 104 are engaged, ring gear 88 and carrier 92 rotate in the opposite direction as input shaft 10. When brakes 102 and 104 are both engaged, ring gear 88 and carrier 92 are held against rotation. Engaging clutch 106 in combination with brake 102 or brake 104 results in two underdrive ratios. Finally, when clutches 100 and 106 are both engaged, ring gear 88 and carrier 92 rotate in unison with input shaft 10.

The transmission of FIG. 6 is like the transmission of FIG. 5 except that clutch 106 is deleted and ring gear 98 is selectively coupled to ring gear 48 by clutch 108. Suggested numbers of gear teeth for each planetary gear set is listed in Table 9.

TABLE 9 Ring 88/Sun 86 2.30 Ring 98/Sun 96 1.50 Ring 48/Sun 46 1.50 Ring 58/Sun 56 1.50

As shown in Table 10, engaging the clutches and brakes in combinations of three establishes nine forward speed ratios and one reverse speed ratio between input shaft 10 and output shaft 12. When the gear sets have tooth numbers as indicated in Table 9, the speed ratios have the values indicated in Table 10.

TABLE 10 60 62 100 102 104 108 Ratio Step Rev X X X −3.83 99% 1^(st) X X X 3.87 2^(nd) X X X 2.33 1.66 3^(rd) X X X 1.67 1.40 4^(th) X X (X) 1.27 1.32 5^(th) X X X 1.00 1.27 6^(th) X X X 0.84 1.19 7^(th) X X X 0.73 1.16 8^(th) X X X 0.60 1.21 9^(th) X X X 0.51 1.17

The transmission of FIG. 7 is like the transmissions of FIG. 5 and FIG. 6 except that both clutch 106 and 108 are present. Suggested numbers of gear teeth for each planetary gear set is listed in Table 11.

TABLE 11 Ring 88/Sun 86 2.30 Ring 98/Sun 96 1.50 Ring 48/Sun 46 1.50 Ring 58/Sun 56 1.50

As shown in Table 12, engaging the clutches and brakes in combinations of three establishes thirteen forward speed ratios and one reverse speed ratio between input shaft 10 and output shaft 12. When the gear sets have tooth numbers as indicated in Table 11, the speed ratios have the values indicated in Table 12.

TABLE 12 60 62 100 102 104 106 108 Ratio Step Rev X X X −4.17 75%  1^(st) X X X 5.56  2^(nd) X X X 4.00 1.39  3^(rd) X X X 2.78 1.44  4^(th) X X X 2.33 1.19  5^(th) X X X 1.67 1.40  6^(th) X X (X) 1.27 1.32  7^(th) X X X 1.00 1.27  8^(th) X X X 0.84 1.19  9^(th) X X X 0.79 1.06 10^(th) X X X 0.72 1.10 11^(th) X X X 0.68 1.06 12^(th) X X X 0.60 1.14 13^(th) X X X 0.52 1.16

FIG. 8 shows a lever diagram corresponding to the gearing arrangements of FIGS. 5-7. Levers 110, 76, and 78 represent gearing arrangements that impose fixedly linear speed relationships. Lever 110 corresponds to gear sets 80 and 90 and represents the linear speed relationship among four shafts that is imposed by these two gear sets. The first shaft corresponds to the combination of sun gear 86 and sun gear 96. The second shaft corresponds to carrier 82. The third shaft corresponds to the combination of ring gear 88, carrier 92, sun gear 46, and ring gear 58. The fourth shaft corresponds to ring gear 98. As discussed above, many known gearing arrangements impose this speed relationship. Lever 76 corresponds to gear set 40. The fifth shaft corresponds to ring gear 48. Lever 78 corresponds to gear set 50. The sixth shaft corresponds to sun gear 56.

While exemplary embodiments are described above, it is not intended that these embodiments describe all possible forms encompassed by the claims. The words used in the specification are words of description rather than limitation, and it is understood that various changes can be made without departing from the spirit and scope of the disclosure. As previously described, the features of various embodiments can be combined to form further embodiments of the invention that may not be explicitly described or illustrated. While various embodiments could have been described as providing advantages or being preferred over other embodiments or prior art implementations with respect to one or more desired characteristics, those of ordinary skill in the art recognize that one or more features or characteristics can be compromised to achieve desired overall system attributes, which depend on the specific application and implementation. These attributes can include, but are not limited to cost, strength, durability, life cycle cost, marketability, appearance, packaging, size, serviceability, weight, manufacturability, ease of assembly, etc. As such, embodiments described as less desirable than other embodiments or prior art implementations with respect to one or more characteristics are not outside the scope of the disclosure and can be desirable for particular applications. 

What is claimed is:
 1. A transmission comprising: an input shaft, an output shaft, and a third shaft; a first gearing arrangement configured to selectively constrain the third shaft to rotate at various speed ratios relative to the input shaft including a negative ratio, two ratios between zero and one, and one, and to selectively hold the third shaft against rotation; a first simple planetary gear set having a first sun gear fixedly coupled to the third shaft, a first ring gear, a first planet carrier fixedly coupled to the input shaft, and at least one planet gear supported for rotation on the first planet carrier and in continuous meshing engagement with the first sun gear and the first ring gear; a second simple planetary gear set having a second sun gear, a second ring gear fixedly coupled to the third shaft, a second planet carrier fixedly coupled to the output shaft, and at least one planet gear supported for rotation on the second planet carrier and in continuous meshing engagement with the second sun gear and the second ring gear; a first brake configured to selectively hold the second sun gear against rotation; and a first clutch configured to selectively couple the output shaft to the first ring gear.
 2. The transmission of claim 1 wherein the first gearing arrangement comprises: first, second, and fourth shafts; a second gearing arrangement configured to fixedly impose a linear speed relationship among the first shaft, the second shaft, the third shaft, and the fourth shaft; a second clutch configured to selectively couple the input shaft to the first shaft; a third clutch configured to selectively couple the input shaft to the second shaft; a second brake configured to selectively hold the second shaft against rotation; and a third brake configured to selectively hold the fourth shaft against rotation.
 3. The transmission of claim 2 further comprising a fourth clutch configured to selectively couple the first shaft to the first ring gear.
 4. The transmission of claim 2 wherein the second gearing arrangement comprises: a third simple planetary gear set having a third sun gear fixedly coupled to the fourth shaft, a third ring gear fixedly coupled to the second shaft, a third planet carrier fixedly coupled to the third shaft, and at least one planet gear supported for rotation on the third planet carrier and in continuous meshing engagement with the third sun gear and the third ring gear; and a fourth simple planetary gear set having a fourth sun gear fixedly coupled to the first shaft, a fourth ring gear fixedly coupled to the fourth shaft, a fourth planet carrier fixedly coupled to the third shaft, and at least one planet gear supported for rotation on the fourth planet carrier and in continuous meshing engagement with the fourth sun gear and the fourth ring gear.
 5. The transmission of claim 4 further comprising a fourth clutch configured to selectively couple the first shaft to the first ring gear.
 6. The transmission of claim 1 wherein the first gearing arrangement comprises: first, second, and fourth shafts; a second gearing arrangement configured to fixedly impose a linear speed relationship among the first shaft, the second shaft, the third shaft, and the fourth shaft; a second clutch configured to selectively couple the input shaft to the first shaft; a third clutch configured to selectively couple the input shaft to the fourth shaft; a second brake configured to selectively hold the first shaft against rotation; and a third brake configured to selectively hold the second shaft against rotation.
 7. The transmission of claim 6 further comprising a fourth clutch configured to selectively couple the fourth shaft to the first ring gear.
 8. The transmission of claim 6 wherein the second gearing arrangement comprises: a third simple planetary gear set having a third sun gear fixedly coupled to the first shaft, a third ring gear fixedly coupled to the third shaft, a third planet carrier fixedly coupled to the second shaft, and at least one planet gear supported for rotation on the third planet carrier and in continuous meshing engagement with the third sun gear and the third ring gear; and a fourth simple planetary gear set having a fourth sun gear fixedly coupled to the first shaft, a fourth ring gear fixedly coupled to the fourth shaft, a fourth planet carrier fixedly coupled to the third shaft, and at least one planet gear supported for rotation on the fourth planet carrier and in continuous meshing engagement with the fourth sun gear and the fourth ring gear.
 9. The transmission of claim 8 further comprising a fourth clutch configured to selectively couple the fourth shaft to the first ring gear.
 10. The transmission of claim 1 wherein the first gearing arrangement comprises: second and fourth shafts; a third gearing arrangement configured to fixedly impose a linear speed relationship among the second shaft, the third shaft, and the fourth shaft; a fourth gearing arrangement configured to selectively impose a linear speed relationship among the input shaft, the third shaft, and the fourth shaft; a third clutch configured to selectively couple the input shaft to the second shaft; a second brake configured to selectively hold the second shaft against rotation; and a third brake configured to selectively hold the fourth shaft against rotation.
 11. The transmission of claim 1 wherein the first gearing arrangement comprises: a first shaft; a third gearing arrangement configured to selectively constrain the first shaft and the third shaft to rotate in opposite directions; a fourth gearing arrangement configured to selectively impose a linear speed relationship among the input shaft, the third shaft, and the first shaft; a second clutch configured to selectively couple the input shaft to the first shaft; and a second brake configured to selectively hold the first shaft against rotation.
 12. A transmission comprising: an input shaft, an output shaft, and first, second, third, fourth, fifth, and sixth shafts; a first gearing arrangement configured to fixedly impose a linear speed relationship among the first shaft, the second shaft, the third shaft, and the fourth shaft; a second gearing arrangement configured to fixedly impose a linear speed relationship among the third shaft, the input shaft, and the fifth shaft; a third gearing arrangement configured to fixedly impose a linear speed relationship among the third shaft, the output shaft, and the sixth shaft; a first clutch configured to selectively couple the output shaft to the fifth shaft; a third clutch configured to selectively couple the input shaft to the second shaft; a first brake configured to selectively hold the sixth shaft against rotation; a second brake configured to selectively hold the second shaft against rotation; and a third brake configured to selectively hold the fourth shaft against rotation.
 13. The transmission of claim 12 further comprising a second clutch configured to selectively couple the first shaft to the input shaft.
 14. The transmission of claim 13 further comprising a fourth clutch configured to selectively couple the first shaft to the fifth shaft.
 15. The transmission of claim 12 further comprising a fourth clutch configured to selectively couple the first shaft to the fifth shaft.
 16. The transmission of claim 12 wherein the first gearing arrangement comprises: a third simple planetary gear set having a third sun gear fixedly coupled to the fourth shaft, a third ring gear fixedly coupled to the second shaft, a third planet carrier fixedly coupled to the third shaft, and at least one planet gear supported for rotation on the third planet carrier and in continuous meshing engagement with the third sun gear and the third ring gear; and a fourth simple planetary gear set having a fourth sun gear fixedly coupled to the first shaft, a fourth ring gear fixedly coupled to the fourth shaft, a fourth planet carrier fixedly coupled to the third shaft, and at least one planet gear supported for rotation on the fourth planet carrier and in continuous meshing engagement with the fourth sun gear and the fourth ring gear.
 17. A transmission comprising: an input shaft, an output shaft, and first, second, third, fourth, fifth, and sixth shafts; a first gearing arrangement configured to fixedly impose a linear speed relationship among the first shaft, the second shaft, the third shaft, and the fourth shaft; a second gearing arrangement configured to fixedly impose a linear speed relationship among the third shaft, the input shaft, and the fifth shaft; a third gearing arrangement configured to fixedly impose a linear speed relationship among the third shaft, the output shaft, and the sixth shaft; a first clutch configured to selectively couple the output shaft to the fifth shaft; a second clutch configured to selectively couple the input shaft to the first shaft; a first brake configured to selectively hold the sixth shaft against rotation; a second brake configured to selectively hold the first shaft against rotation; and a third brake configured to selectively hold the second shaft against rotation.
 18. The transmission of claim 17 further comprising a third clutch configured to selectively couple the fourth shaft to the input shaft.
 19. The transmission of claim 18 further comprising a fourth clutch configured to selectively couple the fourth shaft to the fifth shaft.
 20. The transmission of claim 17 further comprising a fourth clutch configured to selectively couple the fourth shaft to the fifth shaft.
 21. The transmission of claim 17 wherein the first gearing arrangement comprises: a third simple planetary gear set having a third sun gear fixedly coupled to the first shaft, a third ring gear fixedly coupled to the third shaft, a third planet carrier fixedly coupled to the second shaft, and at least one planet gear supported for rotation on the third planet carrier and in continuous meshing engagement with the third sun gear and the third ring gear; and a fourth simple planetary gear set having a fourth sun gear fixedly coupled to the first shaft, a fourth ring gear fixedly coupled to the fourth shaft, a fourth planet carrier fixedly coupled to the third shaft, and at least one planet gear supported for rotation on the fourth planet carrier and in continuous meshing engagement with the fourth sun gear and the fourth ring gear.
 22. A transmission comprising: second, third, fourth, fifth, and sixth shafts; an input shaft; an output shaft; a first gearing arrangement configured to fixedly impose a linear speed relationship among the second shaft, the third shaft, and the fourth shaft; a second gearing arrangement configured to fixedly impose a linear speed relationship among the third shaft, the input shaft, and the fifth shaft; a third gearing arrangement configured to fixedly impose a linear speed relationship among the third shaft, the output shaft, and the sixth shaft; a first clutch configured to selectively couple the output shaft to the fifth shaft; a third clutch configured to selectively couple the input shaft to the second shaft; a first brake configured to selectively hold the sixth shaft against rotation; a second brake configured to selectively hold the second shaft against rotation; and a third brake configured to selectively hold the fourth shaft against rotation.
 23. The transmission of claim 22 further comprising a fourth gearing arrangement configured to selectively impose a linear speed relationship among the input shaft, the third shaft, and the fourth shaft.
 24. The transmission of claim 22 further comprising a fourth gearing arrangement configured to selectively impose a linear speed relationship among the fifth shaft, the third shaft, and the fourth shaft.
 25. A transmission comprising: first, third, fifth, and sixth shafts; an input shaft; an output shaft; a first gearing arrangement configured to selectively constrain the first shaft and the third shaft to rotate in opposite directions; a second gearing arrangement configured to fixedly impose a linear speed relationship among the third shaft, the input shaft, and the fifth shaft; a third gearing arrangement configured to fixedly impose a linear speed relationship among the third shaft, the output shaft, and the sixth shaft; a first clutch configured to selectively couple the output shaft to the fifth shaft; a second clutch configured to selectively couple the input shaft to the first shaft; a first brake configured to selectively hold the sixth shaft against rotation; and a second brake configured to selectively hold the first shaft against rotation.
 26. The transmission of claim 25 further comprising a fourth gearing arrangement configured to selectively impose a linear speed relationship among the input shaft, the third shaft, and the first shaft.
 27. The transmission of claim 25 further comprising a fourth gearing arrangement configured to selectively impose a linear speed relationship among the fifth shaft, the third shaft, and the first shaft. 